Precision surface measurement

ABSTRACT

A test surface of a test object is measured with respect to a reference surface to generate a first relative surface measurement, where the test surface is in a first position relative to the reference surface. The test surface is measured with respect to the reference surface to generate a second relative surface measurement, where the test surface is in a second position relative to the reference surface different from the first position. Estimates of a rotationally varying part of a measurement of the test surface and a rotationally varying part of a measurement of the reference surface are provided. An estimate of a rotationally invariant part of the measurement of the test surface is calculated at a plurality of radial values based on a combination of the relative surface measurements, the provided estimates, and a difference between the first and second relative positions.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Application Ser.No. 60/542,792, filed on Feb. 6, 2004, entitled “METHOD FOR SEPARATINGROTATIONALLY INVARIANT ERRORS IN A PART FROM THOSE IN A MEASUREMENTINSTRUMENT,” incorporated herein by reference.

BACKGROUND

The invention relates to precision surface measurement, includingtechniques to measure surface characteristics of precision opticalcomponents unbiased by a measurement system.

The need to determine the errors in a part (e.g., an optical component,such as a lens or mirror), unbiased by errors in a measurement system,is a long-standing problem. One general approach is to measure multipleobjects in different combinations and then to determine thecontributions from each object individually, such as in the 3-Flat test.The 3-Flat test has a variety of limitations; one of the mostsignificant is the need for three, nominally identical parts to test.

The second general approach is to perform a measurement of the surfaceof a “test part” in a measurement system, and then to displace or“shear” the test part relative to the measurement system. Differencingthe two measurements cancels the contribution (bias) of the measurementsystem and leaves one with an approximation of the derivative of thesurface in the same direction as the shear motion. An estimate of thesurface of the part under test unbiased by the measurement system errormay be derived from the difference data.

Within the family of shearing methods, there are two general approachesthat have been used: lateral shear and rotational shear. Lateral shearin two orthogonal directions results in an estimate of the gradient ofthe part under test; however, the use of lateral shear alone can besensitive to drift resulting in errors proportional to the array size inthe estimate of the test part. Rotational shearing methods have beenwidely reported and have been shown to be robust for the determinationof the rotationally varying surface errors; however, rotational shearingmethods alone do not typically allow one to determine the rotationallyinvariant errors (e.g., mean radial profile) of the test part andinstrument separately from one another.

SUMMARY

Among other aspects, the invention features precision surfacemeasurement techniques including the precision measurement of opticalsurface figure error, and deals in particular with the need to separatethe errors present in a measurement system from errors in the part undertest. For example, the techniques are applicable to any surface shapemeasurement technology where the errors in the measurement device aresignificant compared to the errors in the part under test. One exampleof a surface shape measurement technology is surface profilinginterferometry. Accordingly, the techniques described herein may beapplied to data acquired with surface profiling interferometers,including commercially-available interferometry systems, such as theNewView 5000 available from Zygo (Middlefield, Conn.), and to dataacquired with surface form or wavefront measuring interferometrysystems, such as the GPI family of products available from Zygo(Middlefield, Conn.). The techniques described herein may also beapplied to data acquired with other surface shape measurementtechnologies including non-interferometric technologies including, forexample, data acquired with a coordinate measuring machine (CMM) such asthe PRISMO family of products available from Carl Zeiss, Inc, or dataacquired with a stylus profilometer.

In part, the invention is based on the realization that a rotationallyinvariant part of a test surface measurement (e.g., the mean radialprofile), unbiased by the errors in a reference surface, can be obtainedbased on estimates of the rotationally varying parts of measurements ofthe test and reference surfaces and at least two surface measurements inwhich the test and reference surfaces are laterally displaced (sheared)by a known amount between the two measurements.

We now summarize various aspects and features of the invention.

In one aspect, in general, the invention features a method that includesmeasuring a test surface of a test object with respect to a referencesurface to generate a first relative surface measurement, where the testsurface is in a first position relative to the reference surface. Thetest surface is measured with respect to the reference surface togenerate a second relative surface measurement, where the test surfaceis in a second position relative to the reference surface different fromthe first position. Estimates of a rotationally varying part of ameasurement of the test surface and a rotationally varying part of ameasurement of the reference surface are provided. An estimate of arotationally invariant part of the measurement of the test surface iscalculated at a plurality of radial values based on a combination of thefirst relative surface measurement, the second relative surfacemeasurement, the estimate of the rotationally varying part of themeasurement of the test surface, the estimate of the rotationallyvarying part of the measurement of the reference surface, and adifference between the first and second relative positions.

In another aspect, in general, the invention features a computerreadable medium including a program that causes a processor to receive afirst relative surface measurement of a test surface of a test objectwith respect to a reference surface, where the test surface is in afirst position relative to the reference surface. The processor receivesa second relative surface measurement of the test surface with respectto the reference surface, where the test surface is in a second positionrelative to the reference surface different from the first position. Theprocessor receives estimates of a rotationally varying part of ameasurement of the test surface and a rotationally varying part of ameasurement of the reference surface. The processor calculates anestimate of a rotationally invariant part of the measurement of the testsurface at a plurality of radial values based on a combination of thefirst relative surface measurement, the second relative surfacemeasurement, the estimate of the rotationally varying part of themeasurement of the test surface, the estimate of the rotationallyvarying part of the measurement of the reference surface, and adifference between the first and second relative positions.

In another aspect, in general, the invention features an apparatusincluding an interferometer configured to measure a test surface of atest object with respect to a reference surface to generate a firstrelative surface measurement, where the interferometer includes a stageto position the test surface is in a first position relative to thereference surface. The interferometer is configured to measure the testsurface with respect to the reference surface to generate a secondrelative surface measurement in which the stage is configured toposition the test surface in a second position relative to the referencesurface different from the first position. The apparatus includes anelectronic processor configured to receive the first relative surfacemeasurement and the second relative surface measurement; receiveestimates of a rotationally varying part of a measurement of the testsurface and a rotationally varying part of a measurement of thereference surface; and calculate an estimate of a rotationally invariantpart of the measurement of the test surface at a plurality of radialvalues based on a combination of the first relative surface measurement,the second relative surface measurement, the estimate of therotationally varying part of the measurement of the test surface, theestimate of the rotationally varying part of the measurement of thereference surface, and a difference between the first and secondrelative positions.

Implementations of the invention may include one or more of thefeatures.

The second position includes a laterally displaced position with respectto the reference surface.

The second position of the test surface relative to the referencesurface is achieved by moving the test surface and keeping the referencesurface fixed.

The second position of the test surface relative to the referencesurface is achieved by moving the reference surface and keeping the testsurface fixed.

The estimates of the rotationally varying part of the measurement of thetest surface and the rotationally varying part of the measurement of thereference surface are also determined. For example, determining theestimates of the rotationally varying part of the measurement of thetest surface and the rotationally varying part of the measurement of thereference surface can include measuring the test surface of the testobject with respect to a reference surface to generate a third relativesurface measurement; and measuring the test surface with respect to thereference surface to generate a fourth relative surface measurement,where the test surface is in a rotated position with respect to thereference surface.

The combination is a linear combination.

The rotationally invariant part of the measurement of the test surfaceincludes a mean radial profile of the test surface.

The combination includes a combination of elements of a difference arraybased on the first relative surface measurement, the second relativesurface measurement, the estimate of the rotationally varying part ofthe measurement of the test surface, and the estimate of therotationally varying part of the measurement of the reference surface.

The combination includes a combination of elements of a differencebetween an array representing a difference between the first relativesurface measurement and the second relative surface measurement, and anarray representing a difference between a shifted version of theestimate of the rotationally varying part of the measurement of the testsurface and the estimate of the rotationally varying part of themeasurement of the reference surface.

The combination includes f=Ag, where f is a vector whose elementsinclude values of the mean radial profile, g is a vector whose elementsinclude elements of a difference arrayD(x,y)=M₂(x,y)−M₁(x,y)−[T_(θ)(x−Δx,y−Δy)−T_(θ)(x,y)], T_(θ)(x,y) is theestimate of the rotationally varying part of the measurement of the testsurface, R_(θ)(x,y) is the estimate of the rotationally varying part ofthe measurement of the reference surface, M₁(x,y) is the first relativesurface measurement, M₂(x,y) is the second relative surface measurement,(Δx,Δy) are coordinates of the displacement between the first and secondpositions, and A is a matrix whose elements are calculated based onradial values r=√{square root over (x²+y²)}, and r′=√{square root over((x−Δx)²+(y−Δy)²)}{square root over ((x−Δx)²+(y−Δy)²)}.

In some implementations, A=inv(H^(T)H)H^(T), where inv(H^(T)H) is aninverse or pseudo-inverse of H^(T)H and H is a matrix, each row of whichrepresents an equation that relates the vector f to an element of g.

At least some of the rows of H are determined based on interpolationamong a plurality of elements of vector f.

A row is added to H to prevent H from being singular.

At least some of the plurality of radial values are selected to be moredensely spaced than the size of a pixel with which the test surface wasmeasured.

The plurality of radial values are selected to be more densely spaced asthe radial values increase in magnitude.

In some implementations, measuring the test surface includesinterferometrically measuring the test surface.

The estimate of the rotationally invariant part of the measurement ofthe test surface and the estimate of the rotationally variant part ofthe measurement of the test surface are combined to form a surfaceheight map of the test surface.

The surface height map is expressed in Cartesian coordinates.

The surface height map is expressed in polar coordinates.

The estimate of a rotationally invariant part of the measurement of thereference surface is calculated at a plurality of radial values based onthe estimate of the rotationally invariant part of the measurement ofthe test surface.

The estimate of the rotationally invariant part of the measurement ofthe reference surface is stored to increase the accuracy of subsequentmeasurements of other test surfaces with respect to the referencesurface.

Implementations of the invention may include one or more of thefollowing advantages.

The techniques allow for the separation of errors between test part andmeasurement instrument while suppressing drift errors that othertechniques amplify. The techniques can be performed using only a singletest part and single measurement instrument. The techniques can be usedto measure flats, spheres, and aspheres. The techniques can obtainestimates of the rotationally invariant part of a surface measurement(e.g., mean radial profile) at a sampling density (resolution) greaterthan camera pixel density. Sub-pixel resolution is possible because thesurface is sampled at many combinations of horizontal and vertical pixellocations whose radial distance is not at an integer multiple of thepixel distance.

The techniques can allow for a variety of ways to determine therotationally varying part of the surface measurement.

Computation time and memory storage resources are within thecapabilities of conventional computer systems, even for surfacesmeasured with high resolution.

For example, one aspect that can increase computational efficiency isrepresenting the rotationally invariant surface in terms of the radialcoordinate alone, dramatically reducing the number of unknowns that mustbe solved. For a sense of scale, given a 2000 by 2000 pixel detectorarea and data collected over an inscribed 1000 pixel radius circle, thematrix used in the system of equations would be about 3,140,000 rows(with one row per pixel in the circle) by 3,140,000 columns (with onecolumn per pixel in the circle) if the desired rotationally invariantsurface is calculated over a two-dimensional grid. However, thedimensions of the matrix are 3,140,000 rows by 1,000 columns if therotationally invariant surface is calculated in terms of the radialcoordinate alone at pixel resolution.

The matrix written in terms of the radial coordinate in some cases maystill be too large to process in all but the largest computers. Howeverif we make use of the normal equations and perform matrixmultiplications using outer-products rather than inner-products as isusually done, the matrix that is solved is 1,000 rows by 1,000 columnsand the peak memory usage is related to the small resultant matrix, notthe large matrix, and the problem easily fits into memory. If sub-pixelresolution is employed, the final matrix may be as large as 5,000 rowsby 5,000 columns which still may be solved on readily available but wellequipped desktop computers.

Another aspect that increases computational efficiency is that each rowof the large matrix, which is used to calculate the small matrix, issparse in a way that makes it easy and fast to compute the resultantmatrix. Typically, there are only four or perhaps six non-zero elementsin each row of the large matrix and all of the multiplications andadditions with zero elements can be skipped, which is the vast majorityof operations.

Unless otherwise defined, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which this invention belongs. All publications, patentapplications, patents, and other references mentioned herein areincorporated by reference in their entirety. In case of conflict, thepresent specification, including definitions, will control. In addition,the materials, methods, and examples are illustrative only and notintended to be limiting.

Other features and advantages of the invention will be apparent from thefollowing detailed description, and from the claims.

DESCRIPTION OF DRAWINGS

FIG. 1 is a schematic diagram of a surface form or wavefront measuringinterferometry system.

FIGS. 2A–2C are diagrams of coordinate systems based on pixel locationsfor measured surface data.

FIG. 3 is a flowchart of a surface measurement process.

DETAILED DESCRIPTION

Overview

One example of a system that can be used to measure the surface of atest part is a surface form (or wavefront) interferometry system 100(FIG. 1) in which an interferometer (in this example, a Fizeauinterferometer) is used to combine a test wavefront reflected from thetest surface with a reference wavefront reflected from a referencesurface to form an optical interference pattern or fringe pattern. (By“test surface” we mean to refer to the surface of the test part underinspection.) A fringe pattern is a contour map of the height differencebetween the test and reference surfaces, which is proportional to thephase difference of the light reflected by the reference surface and thelight reflected by the test surface.

Phase-shifting interferometry (PSI) can be used to accurately determinethe phase differences and the corresponding height map of the testsurface. However, the test surface is measured relative to the referencesurface, which may not be error free (e.g., flat) within the tolerancesof the measurement. For many applications (e.g., certification ofoptical components, calibrating a transfer standard, etc.), one desiresan “unbiased measurement” of the test surface, i.e., a surface heightmeasurement that is independent of the reference surface used in theinterferometric measurement (also called “absolute figure metrology”).The techniques described herein can be used to compensate for suchreference surface errors to obtain an unbiased measurement of a testsurface.

Some techniques for obtaining an unbiased measurement of the testsurface include acquiring multiple measurements in which the position,orientation or both of the test surface is changed or “sheared” withrespect to the reference surface from one measurement to the next.Since, as described above, lateral shearing methods for measuring asurface can be sensitive to drift, the system 100 is configured toperform both lateral shearing and rotational shearing of a test surface.The system 100 includes a computer 190 for controlling theinterferometer and for processing data acquired from the lateralshearing and rotational shearing measurements to accurately andefficiently determine an estimate of a mean radial profile, as describedin more detail below. The data processing performed by the computer 190can alternatively be performed using data obtained from any measurementsystem or combination of measurement systems.

Phase Shifting Intedferometry Techniques

With PSI, the optical interference pattern is recorded at multiplespatial locations (or “pixels”) of an image for each of multiplephase-shifts between the reference and test wavefronts to produce, foreach pixel, a series of optical interference patterns that span, forexample, at least a full cycle of optical interference (e.g., fromconstructive, to destructive, and back to constructive interference).The optical interference patterns define a series of irradiance valuesfor each spatial location of the pattern, where each series ofirradiance values has a sinusoidal dependence on the phase-shifts with aphase-offset equal to the phase difference between the combined test andreference wavefronts for that spatial location. Using numericaltechniques known in the art, the phase-offset for each pixel isextracted from the sinusoidal dependence of the irradiance values toprovide a measurement of the test surface relative the referencesurface. Such numerical techniques are generally referred to asphase-shifting algorithms. A two-dimensional surface height measurementcan then be derived from the phase-offset data.

The phase-shifts in PSI can be produced by changing the optical pathlength L₁ from the measurement surface to the interferometer relative tothe optical path length from the reference surface to theinterferometer. For example, in some systems the reference surface ismoved relative to the measurement surface, and in other systems themeasurement surface is moved relative to the reference surface.Alternatively, the phase-shifts can be introduced for a constant,non-zero optical path difference (L₁ not equal to zero) by changing thewavelength of source 140. The latter application is known as wavelengthtuning PSI and is described, e.g., in U.S. Pat. No. 4,594,003 to G. E.Sommargren, the contents of which are incorporated herein by reference.A variety of other techniques for performing PSI can be used including,for example, Fourier Transform Phase Shifting Interferometry (FTPSI) asdescribed in commonly owned U.S. Patent Application Publication No.US-2003-0160968-A1 entitled “PHASE SHIFTING INTERFEROMETRY METHOD ANDSYSTEM” by Leslie Deck, the contents of which are incorporated herein byreference.

Surface Form Interferometry System

Referring to FIG. 1, surface form interferometry system 100 is adaptedto measure the optical interference produced by reflections from acavity 109 formed by a reference surface 121 of a reference optic 120and a test surface 111 of a test object 110. The other surface ofreference optic 120 is typically coated with an anti-reflection layer.Surface 121 is separated from surface 111 by a gap of distance L₁.System 100 includes an adjustable stage 125 under control of thecomputer 190 for positioning test object 110 relative to reference optic120. By controlling the stage 125, the computer 190 is able to rotatethe test object 110 about one or more axes and is able to translate thetest object 110 in any direction. For example, computer 190 controls thestage 125 to laterally shear the test surface 111 along axis 127 togenerate a lateral shear in the image of the test surface 111 at acamera 170. To generate rotational shear, the computer 190 controls thestage 125 to rotate the surface 111 about axis 129. The stage 125 canalso be used to vary the distance L₁ for PSI based on optical pathlength tuning. Alternatively, a separate stage can be used for varyingthe distance L₁.

Depending on the optical imaging properties of the system 100 and theshape of the test surface 111, various combinations of rotation andtranslation of the test object 110 relative to the reference optic 120(while keeping the distance L₁ constant) can yield an effective lateralshear between acquired surface images (i.e., a translation of one imagewith respect to the other), or an effective rotational shear betweenacquired surface images (i.e., a rotation of one image with respect tothe other), or both rotational and lateral shear. For example, togenerate a lateral shear in the image of the surface of a spherical testobject, the stage 125 rotates the spherical test object about itscenter. Alternatively, the system can keep the test object 110 fixed andposition the reference optic 120 relative to the test object 110 toprovide lateral shear, rotational shear, or both.

System 100 additionally includes a tunable light source 140 (e.g., alaser diode), a driver 145 connected to light source 140 for adjustingthe optical frequency of its output, a beam splitter 150, a collimatingoptic 130, an imaging optic 160, a camera 170 (e.g., a CCD camera), anda frame grabber 180 for storing images detected by camera 170. In someembodiments, a single device can perform both control and measurementfunctions (e.g., frame grabber 180 may be incorporated in computer 190).Driver 145 tunes the optical frequency ν of light source 140, through afrequency range Δν about a nominal optical frequency of ν₀.

During operation, computer 190 causes driver 145 to control the opticalfrequency of light emitted by light source 140 and causes frame grabber180 to store an image of the optical interference detected by camera 170for each of the specified optical frequencies. Frame grabber 180 sendseach of the images to computer 190, which analyzes them. In someembodiments, driver 145 linearly modulates the optical frequency of thelight source 140 as the series of interference images are beingrecorded. Alternatively, in other implementations, the driver 145 canmodulate the optical frequency in discrete steps or according to otherfunctions.

During operation, light source 140 directs light having an opticalfrequency ν to beam splitter 150, which then directs the light tocollimating lens 130 to collimate the light into a beam having a nearplanar wavefront. The incoming beam 105 undergoes multiple reflectionsbetween surface 121 and surface 111 of cavity 109. The return beam 107has a wavefront shape that carries information about the optical pathlength. Lenses 130 and 160 image the return beam 107 onto camera 170 toform an optical interference pattern that contains information about anoptical phase φ, which is proportional to the distance L₁ (and thereforerepresents the shape of the test surface 111 relative to the referencesurface 121). Any of a variety of techniques can be used to extract thephase φ, for each pixel to yield a phase distribution φ (x,y) (i.e.,phase map) for the cavity. The result of the phase extraction generatesphases modulo 2π. These phase ambiguities can be accounted for in thephase map using conventional 2π phase ambiguity unwrapping techniques,commonly known in the art. From the phase map, the computer 190generates a surface measurement S(x,y).

Data Processing

In this example, in which the surface measurement S(x,y) is acquired onthe discrete two-dimensional grid of a CCD camera, the function S(x,y)consists of a set of samples of a continuous function {tilde over(S)}(x,y) that represents the height of the test surface relative to theheight of the reference surface. While the function {tilde over(S)}(x,y) is defined for continuous values of x and y over the extent ofthe test surface, the function S(x,y) is defined at a set of discretelocations represented by x and y coordinates corresponding to (e.g., thecentroid of) an area that gets imaged onto a pixel in the plane of thecamera 170. A mapping function (e.g., stored in the memory of computer190) provides the correspondence between a particular pixel address inmemory and the x,y coordinates for the surface measurement recorded atthat pixel. This mapping is able to account for the effects ofdistortion in the imaging optics from the test surface to the camera170.

A typical camera may have millions of pixels, however, for simplifiedillustration, FIG. 2A shows an 11×11 array of 121 pixels. The value ofthe surface measurement z=S(x,y) describing the distance z between thetest surface and the reference surface is based on the total irradianceof a portion of an interference pattern causing charge to build up in apixel of the CCD camera. For the purposes of this example, the discretelocations in the surface measurement S(x,y) measured by the camerapixels will correspond to integral values of x and y.

It is also possible to represent the surface measurement S as a functionof polar coordinates:z=S(x,y)=S(r,θ),where r is the radial distance from the origin (0,0) and θ the anglefrom the +x axis of a point (a pixel center) on the x-y plane. Strictlyspeaking the two forms of S are different in their algebraicrepresentation; however, since they describe the same surface we keepthe same symbol S in the two cases and the coordinate system andrepresentation referenced is implied by the names of the independentvariables.

Furthermore, we note that it is possible to representS(r,θ)=S _(θ)(r,θ)+S _(r)(r),where

${{S_{r}(r)} = {\frac{1}{2\pi}{\int_{0}^{2\pi}{{S\left( {r,\theta} \right)}\ {\mathbb{d}\theta}}}}},$andS _(θ)(r,θ)=S(r,θ)−S _(r)(r).

A function S(r,θ) is the sum of S_(θ)(r,θ), which is referred to as theangularly dependent or rotationally varying (RV) component of thesurface measurement S and S_(r)(r) which is the mean radial profile,angularly independent or rotationally invariant (RI) component of thesurface measurement S. As described above, for many applications onedesires an absolute measurement of the test surface unbiased by thereference surface. However, the surface measurement actually representsthe difference between the test surface and reference surface. The x andy coordinates of the pixel centers can be thought of as indices intoarrays. We can describe a relative surface measurement M as thedifference between the height of a test surface (T) and the height of areference surface (R):M(r,θ)=T(r,θ)−R(r,θ).

There are methods for obtaining T, unbiased by R, by using only lateralshear (see, e.g., Clemens Elster, “Exact two-dimensional wave-frontreconstruction from lateral shearing interferograms with large shears,”Applied Oiptics, 39, No. 29, (2000) pp. 5353–5359); however using onlylateral shear to separate T and R can result in the amplification oferrors due to drift in the test setup. One approach to solving for Tunbiased by R is to write a set of equations relating all of the data toa desired surface characteristic and to perform a least-squares solutionusing a pseudo-inverse technique. The system of equations generatedresults in a matrix whose number of rows and number of columns are equalto the number of pixels in the measurement. Typically, because of thememory constraints, this approach can only use a limited number ofpixels in the calculation due to the size of the resulting matrix.

It is already known that rotational shear may be used to robustlydetermine the RV component of T, unbiased by R (see, e.g., MichaelKüchel, “A new approach to solve the three flat problem,” Optik, 112, No9 (2001) pp. 381–91). The techniques described herein can be used torobustly determine the RI component of T (the mean radial profile)unbiased by R, without amplification of errors due to drift in the testsetup, and using a large number of pixels (or even calculating the meanradial profile with sub-pixel resolution) without demanding excessivememory resources. The RV component of T and the RI component of T canthen be combined to form a surface height map of the test surface,unbiased by the reference surface.

Referring to FIG. 3, a flowchart for a surface measurement process 300includes steps performed by computer 190 to process data obtained by thesystem 100 (or by any other suitable system) to generate the mean radialprofile of the test surface T unbiased by the reference surface R.

Estimates of T_(θ)(x,y) and R_(θ)(x,y), the RV components of T and R,are determined through any of a variety of known techniques including,for example, using the system 100 to perform the rotational shearingmeasurements. The resulting RV components of T and R are provided to(302) (i.e. stored in) computer 190. The system 100 measures (304)surface T with respect to reference surface R to produce a measurementarray M₁, representing a relative surface measurement. Computer 190 thencontrols the stage 125 to move the test object 110 relative to thereference optic 120 to laterally displace (306) test surface T relativeto reference surface R (by an amount that is known or otherwisedetermined) in the image (pixel) coordinate system of the camera 170.The system 100 then performs (308) the resulting laterally shearedmeasurement of T with respect to R to produce a measurement array M₂.

The following data has been acquired and stored in the computer 190 inthe discretely sampled x-y coordinate system:T_(θ)(x,y),R_(θ)(x,y),M ₁(x,y)=T(x,y)−R(x,y), andM ₂(x,y)=T(x−Δx,y−Δy)−R(x,y).

The lateral shear vector between the two measurement positions is(Δx,Δy), corresponding to a lateral shear distance of √{square root over(Δx²+Δy²)}.

Even though the data in this example includes a measurement array M₂that represents the test surface T sheared with respect to its positionin measurement array M₁, it is possible to use data from a relativesurface measurement in which the reference surface R has been shearedwith respect to its position in measurement array M₁:{tilde over (M)} ₂(x,y)=T(x,y)−R(x+Δx,y+Δy)and shift the data such thatM ₂(x,y)={tilde over (M)}₂(x−Δx,y−Δy).

It is also possible to determine the mean radial profile of thereference surface R by subtracting the determined mean radial profile ofthe test surface T from a mean radial profile obtained from themeasurement array M₁. Such a measurement of the mean radial profile ofthe reference surface R can be stored in the computer 190 as acalibration of the reference surface for later use by the system 100,for example, to increase the accuracy of measurements of other testobjects without having to recalibrate the reference surface.

From this data, a difference array D is calculated:D(x,y)=M ₂(x,y)−M ₁(x,y)−[T _(θ)(x−Δx,y−Δy)−T _(θ)(x,y)].

Simplifying yieldsD(x,y)=T _(r)(x−Δx,y−Δy)−T _(r)(x,y)=T _(r)(r′)−T _(r)(r),wherer=√{square root over (x²+y²)},andr′=√{square root over ((x−Δx)²+(y−Δy)²)}{square root over((x−Δx)²+(y−Δy)²)}.Also T_(r) having two independent variables is assumed to be expressedin x-y coordinates, while T_(r) having only one independent variables isassumed to expressed in radial coordinate r (which is appropriate sinceT_(r) is independent of θ).

To form the difference array D, the values of T_(θ)(x,y), R_(θ)(x,y),M₁(x,y), and M₂(x,y) are interpolated if necessary when respective pixelcoordinates x,y are not aligned. For example, this can occur when:lateral shear distance is not a multiple of the pixel size, the imagingoptics of system 100 has distortion (e.g., the pixels in the image planeof the camera are regularly spaced, but the corresponding locations onthe test surface are not regularly spaced), or when testing non-flatcomponents and lateral shear is accomplished by a rotation of the testpart about an axis other than the optical axis of the measurementsystem.

Each element of the array D represents the difference in height betweentwo points in the function T_(r)(r), which is the mean radial profilethat we desire. We can rewrite the equation defining D as a linearsystem of equations:g=Hf,

where g is a column vector made up of the elements of D where there isvalid data (i.e. where the regions of valid data in T_(θ)(x,y),R_(θ)(x,y), M₁(x,y), and M₂(x,y) overlap), f is an (unknown) vectorwhose elements represent samples of the desired mean radial profileT_(r)(r), and H is a matrix to be calculated as described below, eachrow of which is an equation that relates the mean radial profileT_(r)(r) to an element of g. The computer 190 determines (310) theelements of g and H solves (312) this linear system of equations todetermine the desired mean radial profile f.

In general terms, each element of g is the difference between two pointsin the function of T_(r)(r). The elements of g can be indexed in termsof an integral index i, as g(i), mapped to D(x,y) in any order as longas a particular value of i corresponds to a particular pair of (x,y)coordinates and a particular pair of (r,θ) coordinates. (Since r′depends on the value of θ, g is not independent of θ.)

For example, if the lateral shear vector is (Δx,Δy)=(2,0) in units ofpixels, and if i=1 corresponds to (x,y)=(0,0) we haveg(1)=D(0,0)=T _(r)(−2,0)−T _(r)(0,0)=T _(r)(2)−T _(r)(0),andthe corresponding row of H whose inner product with the vector f (j)yields g(1) would beH(1,:)={−1,0,1,0, . . . }.

The colon is used to indicate all column indices yielding all of theelements of the indexed row and the ellipsis to indicate that allremaining elements are 0. In the preceding example, the mean radialprofile f(j) has a uniform resolution of 1 pixel (i.e., withsuccessively indexed values of the mean radial profile separated by adistance in r of 1 pixel). In particular, the elements of f(j) areindexed such that f(1) corresponds to T_(r)(0), f(2) corresponds toT_(r)(1), f(3) corresponds to T_(r)(2), and likewise for higher valuesof i. FIG. 2B shows an exemplary plot of samples of the mean radialprofile for a surface with a convex shape at a uniform pixel resolutionof 1 pixel.

The number of rows of H corresponds to the number of elements in g(i),which typically corresponds to a subset of the pixels in the camera 170such as the number of valid elements of D(x,y) whose x-y coordinatesfall within the intersection of a first a circle 205 within the pixelarray of the camera 170 (e.g., a circle capturing most of a circularfield of view defined by the imaging optics of system 100) representingvalid data in measurement array M₁ and a second circle 207 representingvalid data in the laterally sheared measurement array M₂. The number ofvalid elements of D(x,y) may be less than the number of pixels in theintersection of the valid data circles for other reasons (e.g., badpixels). The number of columns of H corresponds to the number of samplesin the mean radial profile f(j) that are desired.

The matrix H defined in this manner will be singular because the averagevalue (or equivalently any particular value on the vertical axis 210such the origin) of the mean radial profile is undefined. This can beexplained, for example, by noting the measured data is related to themean radial profile by subtracting selected values of the mean radialprofile from other values of the mean radial profile, and the averagevalue of the mean radial profile is lost in the result. This problem canbe solved by adding one equation (one row) that sets the average of themean radial profile to zero, although one may specify any similarconstraint such as setting the value at the origin to zero. For example,the vector g(i) is appended with a single element whose value is 0 andthe matrix H has one row appended with all entries equal to one.

Because the difference array D (and therefore g(i)) is calculated fromdata sampled on a square grid in Cartesian coordinates, but theparameter of interest r for calculating the mean radial profile f(j) isthe Euclidean distance, the mean radial profile is calculated at manypositions which may have integral (pixel length) r values, few of whichhave integral x and y values. As a result, each row of H will need tointerpolate between the available sample points represented in g(i) torelate the elements in f(j) to these interpolated sample points atnon-integral values of x and y corresponding to locations at which themean radial profile f(j) is being determined.

For example, if i=2 corresponds to (x,y)=(0,1) theng(2)=D(0,1)=T _(r)(−2,1)−T _(r)(0,1)=T _(r)(√{square root over (5)})−T_(r)(1),and the corresponding row of H would beH(2,:)={0,−1,0.764,0.236,0, . . . }using 1^(st) order (linear) interpolation.

It is possible to use 0^(th) (nearest neighbor), 1^(st) (linear) orhigher order interpolations in the construction of the matrix H. Inpreferred embodiments, at least linear interpolation or higher orderinterpolation is used. Additionally, the lateral shear distance may notbe an integral multiple of the pixel width, resulting in interpolationbeing used to calculate the elements of g.

Successively indexed values of the mean radial profile f(j) can beseparated by a distance in r of less than one pixel for “sub-pixelresolution,” as shown in FIG. 2C. Since there is an increasing densityof unique values of the Euclidean distance r for any pixel as distancefrom the origin increases, sub-pixel interpolation is likely to be moreaccurate for pixels away from the origin. For example, there is onepixel at r=0, four at r=1, four more at r=1.414, and increasingly morepixels at more densely spaced values of r. Additionally, successivelyindexed values of the mean radial profile f(j) do not need to beuniformly spaced. Thus, some implementations generate values of f(j) andtherefore T_(r)(r) with increasing resolution as r increases.

The “radial shear distance” r′−r varies between the lateral sheardistance √{square root over (Δx²+Δy²)} for samples whose direction θ isparallel to the shear direction (the direction of the shear vector), andzero for samples whose direction θ is orthogonal to the shear direction.Therefore elements of D(x,y) whose (x,y) coordinates are on theperpendicular bisector of the initial and laterally sheared positions ofthe origin (0,0) will result in a row of H, that is all zero resultingin a singular matrix. Additionally, elements of D(x,y) near theperpendicular bisector will have radial shear distances near zero andwill be subject to noise and also should not be included. For example,the element at (x,y)=(1,0) from the example above would corresponds to:D(1,0)=T _(r)(−1,0)−T _(r)(1,0)=T _(r)(1)−T _(r)(1).

The mean radial profile T_(r)(r) is represented within the data of g(i)at increasing sampling density as r increases in the two-dimensionalplane. Some care must be exercised in sub-pixel resolution sampling ofthe mean radial profile since near the origin there are few measurementswhile near the periphery of the aperture the sampling is exceedinglydense. If the locations at which f is to be determined are too finelyspecified then the matrix H will again be singular. To avoid producing asingular matrix one needs to ensure that there are samples of the meanradial profile represented within the data of g(i) between the locationsat which one desires to determine the mean radial profile. One exampleof a sequence of locations in r, in pixel units, at which the meanradial profile may be determined is: 0, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5,5, 5.25, 5.5, 5.75, 6, . . . aperture radius in pixels. When solving theequationg=Hf,

it becomes clear that depending upon the size of the arrays M and D thatthe matrix H can be quite large. The arrays M and D typically haveapproximately one element per camera pixel over a typically square area.The vector g typically has approximately one element of valid data perpixel within a (typically circular) imaging aperture inscribed within a(typically square) camera area. The camera area may range fromapproximately 32×32 pixels to 2048×2048 pixels or more. For a 2048×2048detector this corresponds to a vector g that has approximately threemillion elements. The vector f (and the mean radial profile T_(r)(r))will, however, only need approximately one element per pixel (for 1pixel resolution) along a radius of the circular aperture orapproximately 1000 elements for a 2048×2048 element detector.Additionally, the matrix H is quite sparse. In the case of linearinterpolation most rows of H will have only four non-zero elements outof the 1000 elements in a row in this example. One approach to solvingthe linear matrix equation is to solve the equivalent problemH^(T)g=H^(T)Hf.

In this example, the product H^(T)g has about 1000 elements and H^(T)His a square matrix of about 1000 elements on a side. This sized problemis readily solved by software packages such as Matlab by The MathWorkson a typical Windows based computer.

To successfully solve this problem in a short amount of time, one shouldefficiently calculate H^(T)g and H^(T)H because the matrix H is quitelarge. In one approach, these products can be calculated as aninner-product (dot-product) of columns of H with all other columns of Hor g. However, H can be too large too store in computer memory.Additionally, it is rather expensive in computation time to determineall of the elements of a column of H and the columns need to becalculated many times. Thus, this approach for computing the products anelement at a time is not efficient.

Another more efficient approach to calculating H^(T)H is to determine arow of H, which is easy to do, and make full use of it and repeat foreach row. It turns out that the product H^(T)H can be expressed as a sumof all outer-products of each row of H with itself. Furthermore, eachrow of H is so sparse that one only needs to compute the non-zeroproducts and sum them into the product array. A similar operation can beperformed in the computation of H^(T)g. This approach makes it possibleto produce a system of equations whose solution is the mean radialprofile and that make use of all the data present and still fits in thememory constraints of a standard Microsoft Windows based computer.

It is also important to realize that in many applications one may beinterested in much more modest resolution requirements. In these casesit is possible to forgo these memory conservation techniques anddirectly solve g=Hf. For example, if the data fits in a circle with aradius of 100 pixels, then H is approximately 31,000 by 100 elements andtakes about 24 Megabytes of computer storage, which fits within thememory of most current Windows based computers

It is appropriate to use a large lateral shear distance to improve thesignal-to-noise ratio of the measurement process. This is motivated bythe simple observation that one is measuring height change over adistance and the signal is greater for a fixed slope if the lateralshear distance is larger. If one proceeds directly to solve the equationg=Hf, it turns out that elements of D (equivalently g) for which theradial shear distance is zero are included in the computations.

As described above, a radial shear distance of zero means that the twopoints that are entering the difference equation are the same distancefrom the origin. These measurements can be excluded from the computationby simply leaving out an element of g and corresponding row of H. Allelements in D that are on a line perpendicular to the shear directionand through the midpoint between the origin of the mean radial profilein the original and sheared locations have a radial shear distance ofzero. Elements of D that correspond to very small radial shear distanceare likely to be noisy and can also be excluded from the computation ofH^(T)g and H^(T)H or left out of the equation g=Hf.

It is also possible to combine sets of equations for multiple lateralshears, where some rows of H use one lateral shear vector, and otherrows H use a different lateral shear vector. It is important to realizethat the motion producing the shear may be in any direction.Alternatively, rather than combining multiple shear data sets into oneset of equations, one may use each shear data set to calculate anestimate of the mean radial profile and then average all of the meanradial profile estimates to form an improved estimate of the mean radialprofile.

There are a variety of algorithms for solving a system of linearequations. One reasonable choice is to calculate the singular valuedecomposition (SVD) of H^(T)H=USW^(T) (see, e.g., Roger Horn and CharlesJohnson, “Matrix Analysis”, Cambridge University Press, 1985) where Uand W are unitary matrices and S is a diagonal matrix of singularvalues. This allows one to both identify and resolve a rank deficiencyof the matrix via the pseudo-inverse (H^(T)H)^(↑) rather than a strictinverse. The SVD also provides a means for determining if the samplelocations are causing a rank deficiency. The method is sufficientlyrobust that it is possible to determine the mean radial profile reliablyif there are a few excess sample locations or if the equation definingthe average value of the mean radial profile is left out. In general, itis preferable to resolve these sources of rank deficiency in thedefinition of the system of equations when possible rather than in thesolution method. Methods to determine the SVD of a matrix are welldocumented. It should be noted that the matrix H^(T)H is real andsymmetric and therefore the SVD takes a simpler form H^(T)H=USU^(T)[Horn & Johnson, pg. 157g] and the columns of U are both theeigenvectors and singular vectors of H^(T)H. Calculating the SVD of areal symmetric matrix by obtaining the eigenvalues and eigenvectors usesless memory than a general SVD code.

The desired solution is the mean radial profile f(H ^(T) H)^(↑) H ^(T) g=(H ^(T) H)^(↑) H ^(T) Hf=f.

If one knows the shear displacements and coordinates of the data beforethe data are taken one may compute the pseudo-inverse (H^(T)H)^(↑) inadvance and if H is small enough the matrixA=(H ^(T) H)^(↑) H ^(T)

may be computed in advance. If this is the case, then the solutionrequires collection of the data and a single matrix multiplicationf=Ag.

In some cases H may be to be too large to conveniently store in its fullrepresentation. Even though H is very sparse the productA=(H^(T)H)^(↑)H^(T) is not necessarily sparse. As a result, when H islarge the solution will involve two matrix multiplications, the firstbetween a sparse matrix H stored in a computationally efficient mannerand a data vector gg′=H^(T)g;where g′ is a relatively modest length vector, perhaps 1000 elementslong. The final result is then computed byf=(H ^(T) H)^(↑) g′.

There is a significant computational advantage to a test setup thatprecisely repeats the position of the test part so that thepseudo-inverse of the matrix can be computed in advance. However, ifneed be one may compute the matrix uniquely for each set of dataprocessed so that it is not a requirement to have repeatable positioningfrom measurement to measurement, rather it is only necessary to know thelateral shear vector adequately for any given measurement.

Automation, Image Processing, and Software

In preferred embodiments, the computer controls the stage(s) supportingthe test and or reference objects to automatically position them foreach of the different measurements. To facilitate precisely positioningby the stages, the test and/or reference objects can include one or morealignment marks within the field of view of the camera. Image processingtechniques can then be used to determine the precise motion imparted tothe objects by the stage(s) based on the change in position of thevarious alignment marks as seen by the camera.

In any of the embodiments described above, the computer can includehardware, software, or a combination of both to control the othercomponents of the system and to analyze the surface measurements toextract the desired information about the test surface and/or referencesurface. The analysis described above can be implemented in computerprograms using standard programming techniques. Such programs aredesigned to execute on programmable computers each comprising aprocessor, a data storage system (including memory and/or storageelements), at least one input device, at least one output device, suchas a display or printer. The program code is applied to input data(e.g., phase-shifted images from a CCD camera) to perform the functionsdescribed herein and generate information (e.g., the topography of aselected surface), which is applied to one or more output devices. Eachcomputer program can be implemented in a high-level procedural orobject-oriented programming language, or an assembly or machinelanguage. Each such computer program can be stored on a computerreadable storage medium (e.g., CD ROM or magnetic diskette) that whenread by a computer can cause the processor in the computer to performthe analysis described herein.

A number of embodiments of the invention have been described.Nevertheless, it will be understood that various modifications may bemade without departing from the spirit and scope of the invention.

1. A method comprising: measuring a test surface of a test object withrespect to a reference surface to generate a first relative surfacemeasurement, where the test surface is in a first position relative tothe reference surface; measuring the test surface with respect to thereference surface to generate a second relative surface measurement,where the test surface is in a second position relative to the referencesurface different from the first position; providing an estimate of arotationally varying part of a measurement of the test surface and anestimate of a rotationally varying part of a measurement of thereference surface; calculating an estimate of a rotationally invariantpart of the measurement of the test surface at a plurality of radialvalues based on a combination of the first relative surface measurement,the second relative surface measurement, the estimate of therotationally varying part of the measurement of the test surface, theestimate of the rotationally varying part of the measurement of thereference surface, and a quantity representing the change between thefirst and second relative positions; and outputting the estimate of therotationally invariant part of the measurement of the test surface. 2.The method of claim 1, wherein the second position comprises a laterallydisplaced position of the test surface with respect to the referencesurface.
 3. The method of claim 1, wherein the second position of thetest surface relative to the reference surface is achieved by moving thetest surface and keeping the reference surface fixed.
 4. The method ofclaim 1, wherein the second position of the test surface relative to thereference surface is achieved by the reference surface and keeping thetest surface fixed.
 5. The method of claim 1, further comprisingdetermining the estimate of the rotationally varying part of themeasurement of the test surface and the estimate of the rotationallyvarying part of the measurement of the reference surface.
 6. The methodof claim 5, wherein determining the estimate of the rotationally varyingpart of the measurement of the test surface comprises: measuring thetest surface with respect to the reference surface used in the firstrelative surface measurement or with respect to a different referencesurface to generate a third relative surface measurement; and measuringthe test surface with respect to the reference surface used in thirdrelative surface measurement to generate a fourth relative surfacemeasurement, where the test surface is in a rotated position relative toits position in the third relative surface measurement.
 7. The method ofclaim 1, wherein the combination is a linear combination.
 8. The methodof claim 1, wherein the rotationally invariant part of the measurementof the test surface comprises a mean radial profile of the test surface.9. The method of claim 8, wherein the combination comprises acombination of elements of a difference array based on the firstrelative surface measurement, the second relative surface measurement,the estimate of the rotationally varying part of the measurement of thetest surface, and the estimate of the rotationally varying part of themeasurement of the reference surface.
 10. The method of claim 9, whereinthe combination comprises a combination of elements of a differencebetween an array representing a difference between the first relativesurface measurement and the second relative surface measurement, and anarray representing a difference between a shifted version of theestimate of the rotationally varying part of the measurement of the testsurface and the estimate of the rotationally varying part of themeasurement of the reference surface.
 11. The method of claim 10,wherein the combination comprises f=Ag, where f is a vector whoseelements include values of the mean radial profile, g is a vector whoseelements include elements of a difference arrayD(x,y)=M₂(x,y)−M₁(x,y)−[T_(θ)(x−Δx,y−Δy)−T_(θ)(x,y)], T_(θ)(x,y) is theestimate of the rotationally varying part of the measurement of the testsurface, R_(θ)(x,y) is the estimate of the rotationally varying part ofthe measurement of the reference surface, M₁(x,y) is the first relativesurface measurement, M₂(x,y) is the second relative surface measurement,(Δx,Δy) are coordinates of the displacement between the first and secondpositions, and A is a matrix whose elements are calculated based onradial values r=√{square root over (x²+y²)}, and r′=√{square root over((x−Δx)²+(y−Δy)²)}{square root over ((x−Δx)²+(y−Δy)²)}.
 12. The methodof claim 11, wherein A=inv(H^(T)H)H^(T), where inv(H^(T)H) is an inverseor pseudo-inverse of H^(T)H and H is a matrix, each row of whichrepresents an equation that relates the vector f to an element of g. 13.The method of claim 12, wherein at least some of the rows of H aredetermined based on interpolation among a plurality of elements ofvector f.
 14. The method of claim 12, further comprising adding a row toH to prevent H from being singular.
 15. The method of claim 1, whereinat least some of the plurality of radial values are selected to be moredensely spaced than the size of a pixel with which the test surface wasmeasured.
 16. The method of claim 1, wherein the plurality of radialvalues are selected to be more densely spaced as the radial valuesincrease in magnitude.
 17. The method of claim 1, wherein measuring thetest surface comprises interferometrically measuring the test surface.18. The method of claim 1, further comprising combining the estimate ofthe rotationally invariant part of the measurement of the test surfaceand the estimate of the rotationally variant part of the measurement ofthe test surface to form a surface height map of the test surface. 19.The method of claim 18, wherein the surface height map is expressed inCartesian coordinates.
 20. The method of claim 18, wherein the surfaceheight map is expressed in polar coordinates.
 21. The method of claim 1,further comprising calculating an estimate of a rotationally invariantpart of the measurement of the reference surface at a plurality ofradial values based on the estimate of the rotationally invariant partof the measurement of the test surface.
 22. The method of claim 21,further comprising storing the estimate of the rotationally invariantpart of the measurement of the reference surface to increase theaccuracy of subsequent measurements of other test surfaces with respectto the reference surface.
 23. The method of claim 22, wherein increasingthe accuracy of subsequent measurements of other test surfaces withrespect to the reference surface comprises reducing uncertainty in anestimate of a rotationally invariant part of a subsequent measurement.24. The method of claim 1, wherein outputting the estimate of therotationally invariant part of the measurement of the test surfacecomprises storing the estimate of the rotationally invariant part of themeasurement of the test surface.
 25. The method of claim 1, whereinoutputting the estimate of the rotationally invariant part of themeasurement of the test surface comprises displaying the estimate of therotationally invariant part of the measurement of the test surface. 26.A computer readable medium comprising a program that causes a processorto: receive a first relative surface measurement of a test surface of atest object with respect to a reference surface, where the test surfaceis in a first position relative to the reference surface; receive asecond relative surface measurement of the test surface with respect tothe reference surface, where the test surface is in a second positionrelative to the reference surface different from the first position;receive an estimate of a rotationally varying part of a measurement ofthe test surface and an estimate of a rotationally varying part of ameasurement of the reference surface; calculate an estimate of arotationally invariant part of the measurement of the test surface at aplurality of radial values based on a combination of the first relativesurface measurement, the second relative surface measurement, theestimate of the rotationally varying part of the measurement of the testsurface, the estimate of the rotationally varying part of themeasurement of the reference surface, and a quantity representing thechange between the first and second relative positions; and output theestimate of the rotationally invariant part of the measurement of thetest surface.
 27. The computer readable medium of claim 26, wherein thesecond position comprises a laterally displaced position of the testsurface with respect to the reference surface.
 28. The computer readablemedium of claim 26, wherein the second position of the test surfacerelative to the reference surface is achieved by moving the test surfaceand keeping the reference surface fixed.
 29. The computer readablemedium of claim 26, wherein the second position of the test surfacerelative to the reference surface is achieved by moving the referencesurface and keeping the test surface fixed.
 30. The computer readablemedium of claim 26, wherein outputting the estimate of the rotationallyinvariant part of the measurement of the test surface comprises storingthe estimate of the rotationally invariant part of the measurement ofthe test surface.
 31. The computer readable medium of claim 26, whereinoutputting the estimate of the rotationally invariant part of themeasurement of the test surface comprises displaying the estimate of therotationally invariant part of the measurement of the test surface. 32.An apparatus comprising: an interferometer configured to measure a testsurface of a test object with respect to a reference surface to generatea first relative surface measurement, where the interferometer includesa stage to position the test surface is in a first position relative tothe reference surface; and measure the test surface with respect to thereference surface to generate a second relative surface measurement inwhich the stage is configured to position the test surface in a secondposition relative to the reference surface different from the firstposition; and an electronic processor configured to receive the firstrelative surface measurement and the second relative surfacemeasurement; receive an estimate of a rotationally varying part of ameasurement of the test surface and an estimate of a rotationallyvarying part of a measurement of the reference surface; and calculate anestimate of a rotationally invariant part of the measurement of the testsurface at a plurality of radial values based on a combination of thefirst relative surface measurement, the second relative surfacemeasurement, the estimate of the rotationally varying part of themeasurement of the test surface, the estimate of the rotationallyvarying part of the measurement of the reference surface, and a quantityrepresenting the change between the first and second relative positions.33. The apparatus of claim 32, wherein the second position comprises alaterally displaced position of the test surface with respect to thereference surface.
 34. The apparatus of claim 32, wherein the stage isconfigured to position the test surface in a second position relative tothe reference surface by moving the test surface and keeping thereference surface fixed.
 35. The apparatus of claim 32, wherein thestage is configured to position the test surface in a second positionrelative to the reference surface by moving the reference surface andkeeping the test surface fixed.